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It is a vector bundle over the space of complex structures on A. A complex structure is also an almost complex structure J : TM + T M ,J 2 = - I d . Let 65 be a small deformation of J , then we have ( J 6J)2 = - I d . This is J6J 6 d d = 0. Let 6J be a deformation of complex structure J . The following is a connection on the Hilbert s p x c vector bundle. + + [Vi! Vj] = lCwi3: [V;, Vj] = 0, [V;, V,] = 0. It is importmit to note thal: 1) 6% preserves holornorphicity. This can be verified by showing that it.

We know that it is a vector bundle of complex structures over the punctured sphere. , zn)}. So we have a trivial vector bundle + Vj, 8 vj, @I ... @ Vj, + Cn - A. The projective flat connection can be explicitly described in this case. Let { I p } be an orthonomal basis of s l ( 2 , C ) with respect t o the CartanKilling form. Let is the projection from I$,@ Vj, @ ... @ Vjv, to The Knizhik-Zamolodchikov equations are: where l3@ :=O,i= 8 zi 5,. , n , whcre @ is a section of the Hilbert space vector bundle.

For each flat connection A we have a covariant derivst'ive d~ = d A . Thc tangent space of M is the space of first cohomology H i A ( C ,E @g ) . M is also a symplectic variety with respect to the syrnplectic form w above and one can easily check t,hat w only depend on dA-cohornology classes. This way we push the symplectic form down t o a syrnplectic form on the syrnplectic quotieat. As we have seen in Chapter One it is pretty easy t o quantize an affine symplectic space. W e use holomorphic quantization here.